MAE_242_Lec1.ppt - Engineering Mechanic-Dynamic

1. MAE 242 Dynamics – Section I Dr. Kostas Sierros

2. Important information • MAE 242 – Section 1: Every Tuesday & Thursday 8:00-9:15 AM • All lectures will take place at 113 MRB-E • E-mail: Kostas.Sierros@mail.wvu.edu • Room: G-19 ESB • Phone: 293-3111 ext 2310 • HELP: Tuesday & Thursday 9:30-11:00 AM and by appointment

3. Text books Engineering Mechanics: Dynamics C. Hibbeler, 11th Edition, Prentice Hall, 2006 …and probably some more…

4. Please make sure… • You revise some maths (i.e. trigonometric identities, derivatives and integrals) • … and some STATICS… UNITS, Vector addition, FBD (Hibbeler Statics: Ch. 1,2 and 5)

5. Revision examples (1)

6. Revision examples (2)

7. Revision examples (3)

8. Kinematics of a particle: Objectives • Concepts such as position, displacement, velocity and acceleration are introduced • Study the motion of particles along a straight line. Graphical representation • Investigation of a particle motion along a curved path. Use of different coordinate systems • Analysis of dependent motion of two particles • Principles of relative motion of two particles. Use of translating axis

9. Lecture 1 • Kinematics of a particle (Chapter 12) • -12.1-12.2

10. Material covered • Kinematics of a particle • - Introduction • Rectilinear kinematics: Continuous motion • How to analyze problems • Problems • Next lecture; Rectilinear kinematics: Erratic motion …and maybe a bit more…

11. Kinematics of a particle: Introduction Important contributors Galileo Galilei, Newton, Euler Mechanics Equilibrium of a body that is at rest/moves with constant velocity Accelerated motion of a body Statics Dynamics • Kinematics: geometric aspects of the motion • Kinetics: Analysis of forces which cause the motion

12. Today’s Objectives • Students should be able to: • Find the kinematic quantities (position, displacement, velocity, and acceleration) of a particle traveling along a straight path (12.2) • Next lecture; Determine position, velocity, and acceleration of a particle using graphs (12.3)

13. The displacement of the particle is defined as its change in position. Vector form: r = r’ - r Scalar form:  s = s’ - s Rectilinear kinematics: Continuous motion A particle travels along a straight-line path defined by the coordinate axis s The POSITION of the particle at any instant, relative to the origin, O, is defined by the position vector r, or the scalar s. Scalar s can be positive or negative. Typical units for r and s are meters (m) or feet (ft). The total distance traveled by the particle, sT, is a positive scalar that represents the total length of the path over which the particle travels.

14. The average velocity of a particle during a time interval t is vavg= r/t The instantaneous velocity is the time-derivative of position. v=dr/dt Speed is the magnitude of velocity: v = ds/dt Average speed is the total distance traveled divided by elapsed time: (vsp)avg = sT/  t Velocity Velocityis a measure of the rate of change in the position of a particle.It is avectorquantity (it hasboth magnitude and direction). The magnitude of the velocity is called speed, with units of m/s or ft/s.

15. Acceleration Acceleration is the rate of change in the velocity of a particle. It is a vector quantity. Typical units are m/s2 or ft/s2. The instantaneous acceleration is the time derivative of velocity. Vector form:a= dv/dt Scalar form: a = dv/dt = d2s/dt2 Acceleration can be positive (speed increasing) or negative (speed decreasing). As the book indicates, the derivative equations for velocity and acceleration can be manipulated to get a ds = v dv

16. v t ò ò = = + dv a dt v v a t yields c o c v o o s t ò ò = = + + ds v dt s s v t (1/2)a t yields o o c 2 s o o v s ò ò = = + v dv a ds v (v ) 2a (s - s ) yields 2 2 c o c o v s o o Constant acceleration The three kinematic equations can be integrated for the special case whenacceleration is constant (a = ac)to obtain very useful equations. A common example of constant acceleration is gravity; i.e., a body freely falling toward earth. In this case, ac = g = 9.81 m/s2 = 32.2 ft/s2 downward. These equations are:

17. Important points • Dynamics: Accelerated motion of bodies • Kinematics: Geometry of motion • Average speed and average velocity • Rectilinear kinematics or straight-line motion • Acceleration is negative when particle is slowing down • α ds = v dv; relation of acceleration, velocity, displacement

18. Analyzing problems in dynamics Coordinate system • Establish a position coordinate S along the path and specify its fixed origin and positive direction • Motion is along a straight line and therefore s, v and α can be represented as algebraic scalars • Use an arrow alongside each kinematic equation in order to indicate positive sense of each scalar • Kinematic equations • If any two of α, v, s and t are related, then a third variable can be obtained using one of the kinematic equations • When performing integration, position and velocity must be known at a given instant (…so the constants or limits can be evaluated) • Some equations must be used only when a is constant

19. Problem solving MUSTS • Read the problem carefully (and read it again) • Physical situation and theory link • Draw diagrams and tabulate problem data • Coordinate system!!! • Solve equations and be careful with units • Be critical. A mass of an aeroplane can not be 50 g • Read the problem carefully

20. Problem 1

21. Problem 2

22. Problem 3